And have been thinking about their relevance to the study of consciousness.

Hume's problem of induction, i.e. that one cannot justify inductive knowledge either deductively or inductively, is summarised quite clearly in Hume's 'Inference from the impression to the idea'. Goodman also provides a clear outline of his new riddle of induction in 'Fact, fiction and forecast' (chapter 3).

Relevance to the study of consciousness:

Induction does not seem to be able to lead to certain knowledge; the problems mentioned above (Hume, Goodman) suggest that the practice of science is full of epistemic uncertainty. If science in general cannot provide us with certainty, what then can be determined in studies of consciousness? Given that consciousness is very difficult (perhaps impossible) to 'observe', inductive claims about this phenomenon are even more spurious than in general science. In short, if scientific knowledge is impossible, then so is scientific knowledge of consciousness.
My goodness. It seems to me that consciousness is the only thing that's actually observable. Maybe you mean other people's consciousness? I think you'd like "The View From Nowhere" by Nagel on this topic (especially if you liked "What is it like to be a bat?").

Induction, despite its shortfalls, is the predominant source of scientific knowledge. For this reason it seems likely that an inductive approach will be necessary in furthering investigations into consciousness. Given that this is the case, how can induction be applied most effectively in the study of consciousness? And what kinds of inductive inference are most valid? For example, in Goodman's new riddle of induction, he emphasises the fact that certain hypotheses lend themselves to inductive analysis, whereas others seem not to. How can we separate 'good' inductive hypotheses from 'bad' inductive hypotheses? (My personal opinion is that we draw these distinctions via 'induction on induction', i.e. we learn from experience which inductions work and which do not (via induction!) - but that doesn't really make it easy to pin down which sorts of induction are best...)

In studying consciousness, which is by its nature very difficult to observe, it seems to me that it would be hard to formulate 'good' inductive hypotheses. What sort of criteria would be most important in formulating a good theory of consciousness? (simplicity, testability, its ability to fit in with other theories, etc.)

How do current theories of consciousness rank against these kinds of criteria?

But I think the main point I wanted to get across, is that Hume and Goodman's problems of induction demonstrate that all theories of consciousness are merely theories - none can be established in certain terms. Fundamentally, it doesn't seem possible to know whether we can trust in a particular theory.